Fractals

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Fractal? What’s That?

A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know that some objects can be regarded to have “fractional” dimension?

You can think of dimension of an object X as the amount of information necessary to specify the position of a point in X. For instance, a block of wood is 3-dimensional because you need three coordinates to specify any point inside.

Fractal

The Fractal ( Cantor set ) has fractional dimension! Why? Well it is at most 1-dimensional, because one coordinate would certainly specify where a point is. However, you can get away with “less”, because the object is self-similar.

In Mathworld Fractals are defined as

an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.
This is a fancy way of saying that a fractal is a geometrical figure that has fractional dimension (non integer) including the same pattern, scaled down and rotated, and repeated over and over.
If you still can’t visualize the aspect ( just like almost every first timers ) just peek here.

Fractal

Fractals are said to have a unique property named self similarity. We see the same image again when we “zoom” in and examine a portion of the original.





How Everything Started

Gaston Julia(1893-1978) was a French mathematician who published a book on the iteration of rational functions in 1918. Before computers, he had to draw the sets of functions by hand. These types of fractals are now called Julia sets. His masterpiece on these sets was published in 1918. His interest apparently was piqued by the 1879 paper by Sir Arthur Cayley called The Newton-Fourier Imaginary Problem.

Broccoli fractal

Benoit Mandelbrot (1924- ) is an emeritus professor at Yale University. He used a computer to explore Julia’s iterated functions, and found a simpler equation that included all the Julia sets. Mandelbrot set is named after him.

Waclaw Sierpinski (1882-1969) was a Polish mathematician. His work predated Mandelbrot’s discovery

of fractals. He is known for the Sierpinski triangle, but there are many other Sierpinski-style fractals.

Coastline fractal

In 1918, Bertrand Russell had recognised a “supreme beauty” within the mathematics of fractals that was then emerging. The idea of self-similar curves was taken further by Paul Pierre Levy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Levy C curve. Georg Cantor also gave examples of subsets of the real line with unusual properties – these Cantor sets are also now recognized as fractals.





Properties of Fractals

The essential and most fascinating property of any fractal is its non integer dimension and complexity. The rule for

Snow flake fractal

creating one is essentially simple – A-Level mathematics no more. But the resulting picture has suprising depth. If you zoom in on any part of a fractal, you find the same amount of detail as before. It does not simplify. You find echoes of larger shapes appearing within smaller parts of the shape.

If you zoom in further, the same thing happens. You never seem to get down to the skeleton of the picture, just detail upon detail. Look here or here for illustrations of this. There are many computer programs available for you to do this yourself.
Fractint is probably the best. It is freely available here.





Fractals in Life, Nature or may be beyond that

Coastline fractal in midwest USA

Nature is full of fractals. From a tiny Broccoli, crystals, peacock tail to clouds, snow flakes and blood vessels, approximate fractals are easily found in nature.

Butterfly Effect


Sea shell, urchin,thunder ightnings are also fractals. The Coastline fractal in midwest USA is a surprising example of them as well.

The Butterfly Effect is most succesfully explained by Fractals.

Even the internet (world wide web) is fractal leading us to great links having medium sized links and furthur.
Learn more about fractal nature of internet here.

Butterfly Effect

According to this the universe isformed in a fractal structure, and our cosmos might be a particle, too.
We might be living in a particle. Such particles as the cosmos may exist innumerably.
And there might be a gigantic universe, and it is not the end of all there is. In fact, it might be another particle in another greater universe.

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Project Euler Problem No.5

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Consider the numbers you have to test:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Break up the composite numbers to prime factors:
1, 2, 3, 2*2, 5, 2*3, 7, 2*2*2, 3*3, 2*5,
11, 2*2*3, 13, 2*7, 3*5, 2*2*2*2, 17, 2*3*3, 19, 2*2*5

For a number to be divisible by each of these groups, they have to show up among the prime factorization of our number.

The smallest number divisible by 1, 2, and 3 is 1*2*3 = 6.
To find the smallest number divisible by 1, 2, 3 and 4, we don’t have to multiply all of them together because there’s some redundancies. (2,2,3) is the smallest set that includes the sets (2), (3) and (2,2). So 2*2*3 = 12 is the smallest number divisible by 2, 3 and 4.

There are no 5s in the set (2,2,3), so the first number divisible by 1 through 5 has to be (2,2,3,5) = 60.

6 is 2*3, and we already have that, so we’ve found the number divisible by 1 through 6. Since 7 is prime, our new number has to be (2,2,3,5,7) = 420 in order to be divisible by everything up through 7.

8 is 2*2*2, so in order for (2,2,2) to be a subset we need to add one more two to get (2,2,2,3,5,7).

Continue on like this to get (2,2,2,3,3,5,7) for 9, which is already set for 10=2*5. 11 is prime, so we have (2,2,2,3,3,5,7,11). We’re all set for 12, because 12=2*2*3 and we have two 2s and a 3. For 13 we need (2, 2, 2, 3, 3, 5, 7, 11, 13). This already includes the factors of 14=2*7 and 15=3*5. For 16 we need an extra 2, to get (2, 2, 2, 2, 3, 3, 5, 7, 11, 13).

Then to 17: (2, 2, 2, 2, 3, 3, 5, 7, 11, 13). 18 is all set. 19 makes it (2, 2, 2, 2, 3, 3, 5, 7, 11, 13, 19). And 20 is all set because we already have (2,2,5).

So the shortest number is
2*2*2*2 * 3*3 * 5*7*11*13*19 = 1,3693,680

Fermat number

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Pierre de Fermat wrote to Marin Mersenne on December 25, 1640 that:
If I can determine the basic reason why
3, 5, 17, 257, 65 537, …,
are prime numbers, I feel that I would find very interesting results, for I have already found marvelous things [along these lines] which I will tell you about later. [Archibald1914].
This is usually taken to be the conjecture that every number of the form is prime. So we call these the Fermat numbers, and when a number of this form is prime, we call it a Fermat prime.
The only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537. A simple heuristic shows that it is likely that these are the only Fermat primes (though many folks like Eisentsein thought otherwise).

In 1732 Euler discovered 641 divides F5. It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number Fn with n greater than 2 has the form k.2n+2+1. In the case of F5 this is 128k+1, so we would try 257 and 641 (129, 385, and 513 are not prime). Now we know that all of the Fermat numbers are composite for the other n less than 31. The quickest way to check a Fermat number for primality (if trial division fails to find a small factor) is to use Pepin’s test.

Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g., by straightedge and compass) if and only if n is a power of two times a product of distinct Fermat primes.

The Fermat numbers are pairwise relatively prime, as can be seen in the following identity:

F0F1F2…..Fn-1 +2 = Fn.
(This gives a simple proof that there are infinitely many primes.)
The quickest way to find out if a Fermat number is prime, is to use Pepin’s test [Pepin77]. It is not yet known if there are infinitely many Fermat primes, but it seems likely that there are not.